How to align two differently rotated objects to be exactly side by side
Since these object's local axes are both aligned to their meshes, you can use standard align and snapping tools.. in 2.8, all in Object Mode:


Create a Custom Transform Orientation from the target (stationary)
object. (Orientation dropdown in the header, '+' with target
selected)
With the source (moving) object selected, Object Menu > Transform > 'Align to Transform Orientation'. Now your objects rotations should be the same.
Move the source object roughly into place, and with Snap (ShiftCtrlTab) set to 'Vertex', 'Closest' , with Align to Rotation switched off, snap the source object into place. 


2.8 has lost a lot of standard keymaps, which makes this harder to describe than in 2.79.